Definition 4.1.1 (K0, non-unital case)

Let be a non-unital -algebra, and consider the split exact sequence associated to the Unitizations of .
Define

Since the functor is split exact, we have that no matter whether the algebra is unital or not, the short exact sequence which arises from the unitization, induces an exact sequence of the k-theory groups.

The map is when is unital, and if it is not unital, then it is the inclusion map.